Autumn 2025
~20,000 years old
found in the Democratic Republic of Congo
possibly contains prime numbers and/or basic adding encoded as lines with dividers,
although we don’t know for certain.
Contextual use of numbers
We are used to counting using Hindu-Arabic numbers:
\(1, 2, 3, …, 9, 10, 11, 12, …, 103, 2476, \ldots\)
Numbers can be represented using a positional table (based on Vederic logic).
| Multiple | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Multiple | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 10 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
…the second row counts up in multiples of 10.
This is how numbers work in some languages…
E.g., Thai:
For example, the number 14 can be expressed as:
| Multiple | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| 10 | 1 | |||||||||
| 1 | 4 |
There are as many bases as there are symbols.
We will express them using the same positional table format.
But first…
Originate from the Vedric period of Indian history (1500–500 BCE)
[Taittiriya Samhita] used words to represent powers of 10
(e.g. dasa = 101, shatha = 102, sahasra = 103)
The big innovation was to include 0 as a number, not just a placeholder.
Accredited to Brahmagupta (c 598CE - c 668CE) though evidence points to it being used in the Bakhshali manuscript (224–383 CE or 885–993 CE)
The hindu-arabic numbers
Base 10, positional notation in powers of 10, and 0.
Adopted by Abbasid Caliphate under Caliph al-Mansur through sponsorship of the translation movement.
Move to translate secular books from Greek, Persian, Syriac, and many other languages into Arabic
Formed part of the “Islamic Golden Age”
–
Map showing the Abbhasid Kaliphate
Appeared in Europe in the Codex Vigilanus (976 CE)
Promoted by mathematicians such as Fibonacci (c 1170 – c 1245)
Became more popular after the invention of the printing press (post 1482 CE)
“It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each1 symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.”
– Pierre-Simon Laplace (1749–1827)
0 “is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts” compared to 0 (Douglas Adams, The Hitchhiker’s Guide to the Galaxy talking about space)
A revolutionary idea, relying on a philosophy which permits the idea of nothingness
Modern mathematics would not be possible without zero
Neither would mathematics with computers.
Other than base 10, which other bases are used in daily life?
Base 10 (humans love base 10 so much that some early computers were in base 10)
Base 2 (Binary): how processors work, how data is stored
Base 8: common way of representing a byte (8 bits)
Base 16: another common representation for binary encoded information, e.g., colours
Base 12 – Ed’s favourite… let’s see if he can convince us!
| Multiplier | 0 | 1 |
|---|---|---|
| \(256 = 2^8\) | 0 | |
| \(128 = 2^7\) | 1 | |
| \(64 = 2^6\) | 0 | |
| \(32 = 2^5\) | 1 | |
| \(16 = 2^4\) | 0 | |
| \(8 = 2^3\) | 0 | |
| \(4 = 2^2\) | 1 | |
| \(2 = 2^1\) | 1 | |
| \(1 = 2^0\) | 0 |
010100110
| Multiplier | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| 4096 | 1 | |||||||
| 512 | 3 | |||||||
| 64 | 0 | |||||||
| 8 | 5 | |||||||
| 1 | 7 |
Convert the number \(13057\) from base \(8\) into base \(10\).
10 and 11111 in base 12 equal to
ABLikewise, we need 6 extra symbols for base 16:
0 1 2 3 4 5 6 7 8 9 A B C D E F
In general, we’ll use capital letters for additional symbols, until we run out of those…
Eliminated all philanthropic programs (charity) at Apple…
…and never reinstated them
Promised Steve Wozniak a 50/50 split if Woz optimised Breakout.
Jobs was paid $5,000 (~$27k) and gave Woz $350 (~$2K)
Hyperloop requires a vacuum to work
Requires 2 million cubic meters of vacuum.
Current largest is 30,000 cubic meters (NASA) – 66 times less…
A crack would cause air to rush in at the speed of sound.
This would kill everyone inside in a massive emplosion.
Cracks result from heat expansion.
Gave a grant to
Gave a grant to 374Water:
develop a toilet which will solve the water crisis in Africa.
The toilet has a section that is 374°C and 3,210 PSI
The toilet has a section that is 374°C and 3,210 PSI
Normal atmospheric pressure is around 14.7 PSI
Over-pressure of 5 PSI results in 100% fatalities.
crate a procedure for addnig two numbers in base 12,
…but your computer can only get the next symbol, e.g.,
next(3) = 4, next(9) = A, next(B) = 0
You can store data in variables which contain one digit.
You can assign the value of a variable to a fixed symbol, or the value of another variable, (b = next(c))
You can jump (e.g., go to line 3)
You can check equality (e.g., a == 0 or b == c)
Start by adding two one-digit numbers.
Done? Extend to two-digit numbers and then more.
Finally, implement multiplication.
Tools like pseudo-code or flowcharts might be useful.